A 2 by X rectangle can obviously have 2*X circles of diameter 1 placed inside it without overlapping. What is the smallest integer X where a 2 by X box can have 2*X+1 non-overlapping circles placed inside it?

Same here. I'm not the greatest with geometry, and maybe that's what's going on, but I feel I must be misunderstanding something about the question, because as I understand it right now, it seems to be quite obviously impossible. :P